We prove a lower bound of ¾ Ò ¾ ¿Ò for the rank of Ò ¢Ò-matrix multiplication over an arbitrary field. Similar bounds hold for the rank of the multiplication in noncommutative division algebras and for the multiplication of upper triangular matrices.
We consider the two-variable interlace polynomial introduced by Arratia, . We develop graph transformations which allow us to derive point-to-point reductions for the interlace polynomial. Exploiting these reductions we obtain new results concerning the computational complexity of evaluating the interlace polynomial at a fixed point. Regarding exact evaluation, we prove that the interlace polynomial is #P-hard to evaluate at every point of the plane, except on one line, where it is trivially polynomial time computable, and four lines, where the complexity is still open. This solves a problem posed by Arratia, . In particular, three specializations of the two-variable interlace polynomial, the vertex-nullity interlace polynomial, the vertex-rank interlace polynomial and the independent set polynomial, are almost everywhere #P-hard to evaluate, too. For the independent set polynomial, our reductions allow us to prove that it is even hard to approximate at every point except at 0. * A preliminary version of this work has appeared in the proceedings of STACS 2008.
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